Superconvergence by 𝑀-decompositions. Part I: General theory for HDG methods for diffusion

Cockburn, Bernardo; Fu, Guosheng; Sayas, Francisco

doi:10.1090/mcom/3140

Superconvergence by $M$-decompositions. Part I: General theory for HDG methods for diffusion
HTML articles powered by AMS MathViewer

by Bernardo Cockburn, Guosheng Fu and Francisco Javier Sayas PDF

Math. Comp. 86 (2017), 1609-1641 Request permission

Abstract:

We introduce the concept of an $M$-decomposition and show how to use it to systematically construct hybridizable discontinuous Galerkin and mixed methods for steady-state diffusion methods with superconvergence properties on unstructured meshes.

References

D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 1, 7–32 (English, with French summary). MR 813687, DOI 10.1051/m2an/1985190100071
Franco Brezzi, Jim Douglas Jr., Michel Fortin, and L. Donatella Marini, Efficient rectangular mixed finite elements in two and three space variables, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 4, 581–604 (English, with French summary). MR 921828, DOI 10.1051/m2an/1987210405811
Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235. MR 799685, DOI 10.1007/BF01389710
Paul Castillo, Bernardo Cockburn, Ilaria Perugia, and Dominik Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. 38 (2000), no. 5, 1676–1706. MR 1813251, DOI 10.1137/S0036142900371003
Brandon Chabaud and Bernardo Cockburn, Uniform-in-time superconvergence of HDG methods for the heat equation, Math. Comp. 81 (2012), no. 277, 107–129. MR 2833489, DOI 10.1090/S0025-5718-2011-02525-1
Z. Chen and J. Douglas Jr., Prismatic mixed finite elements for second order elliptic problems, Calcolo 26 (1989), no. 2-4, 135–148 (1990). MR 1083050, DOI 10.1007/BF02575725
Eric T. Chung and Björn Engquist, Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimensions, SIAM J. Numer. Anal. 47 (2009), no. 5, 3820–3848. MR 2576522, DOI 10.1137/080729062
B. Cockburn, Static condensation, hybridization, and the devising of the HDG methods, in Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, G. R. Barrenechea, F. Brezzi, A. Cagiani, and E. H. Georgoulis, eds., Durham Symposium, London Mathematical Society, Durham, U.K., July 7–16, 2014, Lecture Notes in Computational Science and Engineering, vol. 114, Springer, 2016, pp. 129–177.
Bernardo Cockburn, Daniele A. Di Pietro, and Alexandre Ern, Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods, ESAIM Math. Model. Numer. Anal. 50 (2016), no. 3, 635–650. MR 3507267, DOI 10.1051/m2an/2015051
Bernardo Cockburn, Bo Dong, and Johnny Guzmán, A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems, Math. Comp. 77 (2008), no. 264, 1887–1916. MR 2429868, DOI 10.1090/S0025-5718-08-02123-6
Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319–1365. MR 2485455, DOI 10.1137/070706616
Bernardo Cockburn, Jayadeep Gopalakrishnan, and Francisco-Javier Sayas, A projection-based error analysis of HDG methods, Math. Comp. 79 (2010), no. 271, 1351–1367. MR 2629996, DOI 10.1090/S0025-5718-10-02334-3
Bernardo Cockburn, Johnny Guzmán, and Haiying Wang, Superconvergent discontinuous Galerkin methods for second-order elliptic problems, Math. Comp. 78 (2009), no. 265, 1–24. MR 2448694, DOI 10.1090/S0025-5718-08-02146-7
Bernardo Cockburn, Weifeng Qiu, and Ke Shi, Conditions for superconvergence of HDG methods for second-order elliptic problems, Math. Comp. 81 (2012), no. 279, 1327–1353. MR 2904581, DOI 10.1090/S0025-5718-2011-02550-0
Bernardo Cockburn, Weifeng Qiu, and Ke Shi, Superconvergent HDG methods on isoparametric elements for second-order elliptic problems, SIAM J. Numer. Anal. 50 (2012), no. 3, 1417–1432. MR 2970749, DOI 10.1137/110840790
Bernardo Cockburn and Vincent Quenneville-Bélair, Uniform-in-time superconvergence of the HDG methods for the acoustic wave equation, Math. Comp. 83 (2014), no. 285, 65–85. MR 3120582, DOI 10.1090/S0025-5718-2013-02743-3
Bernardo Cockburn and Francisco-Javier Sayas, Divergence-conforming HDG methods for Stokes flows, Math. Comp. 83 (2014), no. 288, 1571–1598. MR 3194122, DOI 10.1090/S0025-5718-2014-02802-0
Bernardo Cockburn and Ke Shi, Conditions for superconvergence of HDG methods for Stokes flow, Math. Comp. 82 (2013), no. 282, 651–671. MR 3008833, DOI 10.1090/S0025-5718-2012-02644-5
Bernardo Cockburn and Ke Shi, Superconvergent HDG methods for linear elasticity with weakly symmetric stresses, IMA J. Numer. Anal. 33 (2013), no. 3, 747–770. MR 3081483, DOI 10.1093/imanum/drs020
Daniele A. Di Pietro and Alexandre Ern, A hybrid high-order locking-free method for linear elasticity on general meshes, Comput. Methods Appl. Mech. Engrg. 283 (2015), 1–21. MR 3283758, DOI 10.1016/j.cma.2014.09.009
Daniele A. Di Pietro, Alexandre Ern, and Simon Lemaire, An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators, Comput. Methods Appl. Math. 14 (2014), no. 4, 461–472. MR 3259024, DOI 10.1515/cmam-2014-0018
Herbert Egger and Joachim Schöberl, A hybrid mixed discontinuous Galerkin finite-element method for convection-diffusion problems, IMA J. Numer. Anal. 30 (2010), no. 4, 1206–1234. MR 2727822, DOI 10.1093/imanum/drn083
Lucia Gastaldi and Ricardo H. Nochetto, Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations, RAIRO Modél. Math. Anal. Numér. 23 (1989), no. 1, 103–128 (English, with French summary). MR 1015921, DOI 10.1051/m2an/1989230101031
C. Lehrenfeld, Hybrid Discontinuous Galerkin methods for solving incompressible flow problems, Diploma Thesis, Rheinisch-Westfälischen Technischen Hochschule, Aachen, 2010.
Issei Oikawa, A hybridized discontinuous Galerkin method with reduced stabilization, J. Sci. Comput. 65 (2015), no. 1, 327–340. MR 3394448, DOI 10.1007/s10915-014-9962-6
P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR 0483555
Rolf Stenberg, A family of mixed finite elements for the elasticity problem, Numer. Math. 53 (1988), no. 5, 513–538. MR 954768, DOI 10.1007/BF01397550
Rolf Stenberg, Postprocessing schemes for some mixed finite elements, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 1, 151–167 (English, with French summary). MR 1086845, DOI 10.1051/m2an/1991250101511
Lars B. Wahlbin, Superconvergence in Galerkin finite element methods, Lecture Notes in Mathematics, vol. 1605, Springer-Verlag, Berlin, 1995. MR 1439050, DOI 10.1007/BFb0096835